Choice-free Topological Duality for Implicative Lattices and Heyting Algebras

02/07/2023

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by   Chrysafis Hartonas, et al.

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We develop a common semantic framework for the interpretation both of 𝐈𝐏𝐂, the intuitionistic propositional calculus, and of logics weaker than 𝐈𝐏𝐂 (substructural and subintuitionistic logics). This is done by proving a choice-free representation and duality theorem for implicative lattices, which may or may not be distributive. The duality specializes to a choice-free duality for the category of Heyting algebras and a category of topological sorted frames with a ternary sorted relation.